6.双曲线 x^2/n-y2=1(n>1)的两焦点为F1,F2,P在双曲线上,且满足|PF1|+|PF2|=2√(n+2),则△PF1F2的面积为(　　)

A. 1/2 B.1C.2D.4

答案:B

7.一动圆过定点A(-4,0),且与定圆B:(x-4)2+y2=16相外切,则动圆圆心P的轨迹方程为　.

解析:因为动圆圆心P到点A(-4,0)与到点B(4,0)的距离差||PA|-|PB||=4,所以其轨迹是以A,B为焦点的双曲线的左支.

答案:x^2/4-y^2/12=1(x≤-2)

8.已知双曲线的方程为x^2/a^2 -y^2/b^2 =1(a>0,b>0),A,B在双曲线的右支上,线段AB经过双曲线的右焦点F2,|AB|=m,F1为另一焦点,则△ABF1的周长为　　　　　.

解析:由双曲线定义得|AF1|-|AF2|=2a,|BF1|-|BF2|=2a,

　　∴|AF1|+|BF1|-(|AF2|+|BF2|)=4a.

　　∴|AF1|+|BF1|=4a+m.

　　∴△ABF1的周长是4a+2m.

答案:4a+2m

9.求适合下列条件的双曲线的标准方程:

(1)a=2√5,经过点A(2,-5),焦点在y轴上;

(2)过点A(3,2)和B(17,12).

解:(1)因为双曲线的焦点在y轴上,所以可设双曲线的标准方程为y^2/a^2 -x^2/b^2 =1(a>0,b>0).

　　由题设知,a=2√5,且点A(2,-5)在双曲线上,

　　所以{■(a=2√5 "," @25/a^2 "-" 4/b^2 =1"," )┤解得a2=20,b2=16.

　　故所求双曲线的标准方程为y^2/20-x^2/16=1.

　　(2)方法一:若焦点在x轴上,设双曲线的标准方程为x^2/a^2 -y^2/b^2 =1(a>0,b>0).

　　由已知条件得{■(9/a^2 "-" 4/b^2 =1"," @(17^2)/a^2 "-" (12^2)/b^2 =1"," )┤解得{■(1/a^2 =1"," @1/b^2 =2"." )┤

　　则双曲线的标准方程为x2-2y2=1.

　　若焦点在y轴上,设双曲线的标准方程为y^2/a^2 -x^2/b^2 =1(a>0,b>0).

　　由已知条件得{■(4/a^2 "-" 9/b^2 =1"," @(12^2)/a^2 "-" (17^2)/b^2 =1"," )┤

　　解得{■(1/a^2 ="-" 2"," @1/b^2 ="-" 1"," )┤不符合题意,舍去.

　　综上所述,所求双曲线的标准方程为x2-2y2=1.

　　方法二:设双曲线方程为mx2-ny2=1(mn>0),

　　则{■(9m"-" 4n=1"," @17^2 m"-" 12^2 n=1"," )┤解得{■(m=1"," @n=2"." )┤

　　∴双曲线的标准方程为x2-2y2=1.

10.某工程要挖一个横截面为半圆的柱形隧道,挖出的土只能沿道路AP,BP运到P处(如图),|PA|=100 m,|PB|=150 m,∠APB=60°,试说明怎样运土才能最省工.